assimilationofsnowcoveredareainformationintohydrologic...

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ADWR 929 No. of Pages 14

10 November 2005 Disk UsedARTICLE IN PRESS

Advances in Water Resources xxx (2005) xxx–xxx

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Assimilation of snow covered area information into hydrologicand land-surface models

Martyn P. Clark a,*, Andrew G. Slater a, Andrew P. Barrett a, Lauren E. Hay b,Gregory J. McCabe b, Balaji Rajagopalan a, George H. Leavesley b

a Cooperative Institute for Research in Environmental Sciences, Center for Science and Technology Policy Research, University of Colorado,

Boulder, CO 80309-0488, United Statesb United States Geological Survey, Water Resources Discipline, Denver, CO, United States

Received 31 May 2005; received in revised form 29 September 2005; accepted 2 October 2005
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PAbstract

This paper describes a data assimilation method that uses observations of snow covered area (SCA) to update hydrologic modelstates in a mountainous catchment in Colorado. The assimilation method uses SCA information as part of an ensemble Kalmanfilter to alter the sub-basin distribution of snow as well as the basin water balance. This method permits an optimal combinationof model simulations and observations, as well as propagation of information across model states. Sensitivity experiments are con-ducted with a fairly simple snowpack/water-balance model to evaluate effects of the data assimilation scheme on simulations ofstreamflow. The assimilation of SCA information results in minor improvements in the accuracy of streamflow simulations nearthe end of the snowmelt season. The small effect from SCA assimilation is initially surprising. It can be explained both becausea substantial portion of snowmelts before any bare ground is exposed, and because the transition from 100% to 0% snow coverageoccurs fairly quickly. Both of these factors are basin-dependent. Satellite SCA information is expected to be most useful in basinswhere snow cover is ephemeral. The data assimilation strategy presented in this study improved the accuracy of the streamflow sim-ulation, indicating that SCA is a useful source of independent information that can be used as part of an integrated data assimilationstrategy.� 2005 Elsevier Ltd. All rights reserved.

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O1. Introduction

Remotely sensed snow covered area (SCA) informa-tion has been used in hydrologic simulation models ina large number of applications (e.g., [17,18,21–23,31,9,28,3,4,19]). One approach is to use SCA as ahydrologic model input. For example, in the snowmeltrunoff model (SRM) the snowmelt equations are appliedto the fraction of the basin that is covered in snow (e.g.,[18]). In this approach snow water equivalent (SWE) is

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0309-1708/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2005.10.001

* Corresponding author. Tel.: +1 303 735 3624; fax: +1 303 7351576.

E-mail address: [email protected] (M.P. Clark).

not computed directly by the model, but can be inferredfrom snow cover depletion curves [9], facilitatingstreamflow forecasts.

Another approach is to use SCA information toalter the sub-basin distribution of SWE. For example,Turpin et al. [31] replaced modeled SWE with modeledSWE from a different date that had similar fractionalSCA to the SCA in a Landsat image. Barrett [4] andMcguire and Lettenmaier [19] both used rule-basedapproaches to modify the sub-basin SWE distribution.Barrett [4] updated snow in two ways. In grid cellswhere the model estimated snow and the satellite didnot, SWE was either removed from the basin, orSWE was redistributed to areas where the satellite

mailto:[email protected]

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SNOWFALL (Ps)

(Qb)BASEFLOW

(Qd)DIRECTRUNOFF

SUB-GRIDSWE

RAINFALL (Pr )

SNOWMELT (Md)

SMCAPWu

DRAINAGE (D)

fr2gw

Wb

frdir

Wi

EVAPORATION (E )

Fig. 1. Conceptual diagram of the snowpack water-balance modelconfigured for this study. A specified fraction of water inputs(rainfall + snowmelt) is extracted as direct runoff (based on parameterfrdir). The soil moisture reservoir is represented as an imperviousbucket, and drainage only occurs once it has reached capacity (definedby parameter smcap, Eq. (11)). Groundwater storage is infinite, withbaseflow parameterized as a function of the groundwater storage(parameter fr2gw; Eq. (15)). Streamflow is taken as the sum of direct

2 M.P. Clark et al. / Advances in Water Resources xxx (2005) xxx–xxx



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and model agree (generally higher elevations). Barrettadded a thin layer of snow to grid cells where the satel-lite estimated snow and the model did not. Resultsshowed, somewhat predictably, that streamflowdecreased when snow was removed from the basin,and that the peak streamflow was delayed when snowwas added to higher elevations. In most cases snowupdating decreased the skill of streamflow simulations.Mcguire and Lettenmaier [19] used the following rules:if the modeled SWE was zero and the satellite estimateof fractional SCA was greater than 50%, then a thinlayer of snow was added to the grid cell; if the modeledSWE was non-zero and the fractional SCA was lessthan 50%, then snow was removed from the grid cell.Mcguire and Lettenmaier [19] showed that their snowupdating procedures improved short-term (14-day)forecasts in a subset of the basins studied.

There are two main problems with these rule-basedapproaches to snow updating. First, rule-based methodssimply replace model simulations with satellite observa-tions [this is known as the ‘‘direct-insertion’’ approachto data assimilation]. The direct-insertion approachassumes that satellite observations are perfect andimplies that there is no useful information in the modelsimulations of snowpack. A second problem with theserule-based snow updating methods is that it is difficultto propagate information to other model state variables.For example, Mcguire and Lettenmaier [19] assume thatin cases when the model shows snow and the satellitedoes not, that water has ‘‘left’’ the basin. In reality,water may linger in soil and groundwater reservoirsbefore being ‘‘seen’’ as streamflow at the basin outlet.A superior approach is a Kalman-type data assimilationstrategy (e.g., [25,26]) that blends satellite products withmodel simulations so as to exploit the relative strengthsin both the model simulations and satellite observations.Moreover, in the Kalman-type data assimilation meth-ods it is fairly straightforward to update other modelstates (e.g., soil moisture) based on the modeled covari-ance across model states.

The purpose of this paper is to demonstrate howsatellite observations of SCA can be used as part of aKalman-type data assimilation strategy to improvemodel simulations of streamflow. Results are based onsensitivity experiments with a fairly simple snowpack/water-balance model. The remainder of this paper isorganized as follows. The hydrologic model is describedin the next section. The data assimilation strategy isdescribed in Section 3, and the probabilistic approachto model simulation is described in Section 4. Modelcovariance is discussed in Section 5, and data assimila-tion experiments are described in Section 6. The paperconcludes with discussion of the potential utility ofSCA information for model updates in a range of differ-ent snow environments.

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2. Model description

The hydrologic model configured for this study usestemperature index methods to model snow accumula-tion and ablation processes (e.g., [24]). The basin waterbalance and streamflow are modeled using conceptualstorage reservoirs. Fig. 1 presents a conceptual diagramof the model, summarizing the main equations andparameters. The simplicity of this model has someadvantages in that the effects of model parameters andstate updating are easy to understand. The main advan-tage is that large ensemble simulations can be performedwith little computational effort.

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2.1. Snow model

SWE and fractional SCA are modeled using sub-gridparameterizations similar to those described by Luceet al. [16] and Liston [14]. The basic premise is thatthe variability in snow within a grid cell can be describedby a parametric probability distribution function(p.d.f.). Consider the two-parameter lognormal p.d.f.used by Liston [14]:

runoff and baseflow (normalized by basin area).

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M.P. Clark et al. / Advances in Water Resources xxx (2005) xxx–xxx 3



f ðwÞ ¼ 1

wfffiffiffiffiffiffi2p

p exp � 1

2

lnðwÞ � kf

� �2( )ð1aÞ

with parameters

k ¼ lnðlÞ � 1

2f2 ð1bÞ

f ¼ lnð1þ CV2Þ ð1cÞ

where w is snow water equivalent (mm), l is the mean ofthe probability distribution, and CV is the coefficient ofvariation.

Luce et al. [16] and Liston [14] both define the p.d.f. interms of total accumulation, and assume that melt is spa-tially uniform over a grid cell. In this implementation it isonly necessary to model total snow accumulation, Da

(mm), and total melt depth, Dm (mm). Here, Da = l inEq. (1), and is defined as the sum of individual snowfallevents. Likewise, Dm is defined as the sum of individualmelt events. Assuming uniform melt, SCA and SWE areestimated by integrating over the p.d.f. defined by Da

and CV (Eq. (1)), that is

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Fig. 2. Various lognormal probability distribution functions (p.d.f.s) describi(dark lines) and mean = 200 mm (light lines), and coefficient of variation paraof the distribution of SWE associated with each p.d.f.; (c) effects of melt on theparameterization of SWE.

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SCA ¼Z 1

Dm

f ðwÞdw ð2Þ

SWE ¼Z 1

Dm

ðw� DmÞf ðwÞdw ð3Þ

Note that Dm is the lower limit of integration. Physicallyspeaking, these equations amount to identifying areas ofequal SWE amount across a basin (e.g., 3.5% of the ba-sin has SWE of 500 mm; 2.7% of the basin has SWE of510 mm; and so forth), and computing the weighted sumof SWE amounts (weighted by area). SCA is simply thesum of areas with SWE on the ground. Liston [14] pre-sents analytical solutions to Eqs. (2) and (3).

P.d.f.s computed using different CV parameters areillustrated in Fig. 2a, and a conceptual example of theresultant distribution of snow is provided in Fig. 2b(the lines in Fig. 2b are essentially the cumulativep.d.f.s flipped on their side). The effects of melt are illus-trated for the p.d.f.s in Fig. 2c, and for the examplesnow distribution in Fig. 2d. Conceptually, the shallowportions of the grid cell will melt first, reducing the snowcovered area in the grid cell (geometrically, melt causes

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ng sub-grid variability in SWE, showing (a) p.d.f.s for mean = 100 mmmeter = 0.3 (thin lines) and 0.7 (thick lines); (b) conceptual illustrationp.d.f.s; and (d) conceptual illustration of the effects of melt on sub-grid

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the p.d.f. to shift left (Fig. 2c)). Because the sub-grid var-iability in SWE is determined based on total accumula-tion, snow can remain in the grid cell even if total melt isgreater than total accumulation.

As just noted, total snow accumulation Da (mm) isdefined as the sum of precipitation from daily snowfallevents. Precipitation is modeled as snow if the tempera-ture is below a prescribed threshold; otherwise precipita-tion is rain:

P s ¼P � pbias; T d < tsnow

0; T d P tsnow

�ð4aÞ

P r ¼0; T d < tsnow

P � pbias; T d P tsnow

�ð4bÞ

Here P is precipitation (mm d�1), Ps and Pr are precip-itation in the form of snow and rain, respectively(mm d�1), pbias is a dimensionless parameter definingthe bias in precipitation, Td is the mean daily tempera-ture (�C), and tsnow is a model parameter for the tem-perature threshold that determines precipitation phase(�C). Rain percolates through the snowpack without de-lay, and is immediately available to the water-balancemodel.

Snowmelt, Ms (mm d�1), is estimated using tempera-ture index methods described in Rango and Martinec[24]:

M s ¼ðT d � tmeltÞ � ddpar; T d > tmelt

0; T d 6 tmelt

�ð5Þ

As before, Td is the mean daily temperature (�C), andddpar (mm �C�1 d�1) and tmelt (�C) are model parame-ters. In Rango and Martinec�s implementation of thismethod, ddpar is a parameter that increases throughthe melt season to account for seasonal changes in solarradiation, snow temperature, and so forth. Here ddparis estimated based on the mean temperature for the past30 days (T30):

ddpar ¼ jðT 30 � tbaseÞtdpar; T 30 > tbase

j tbase; T 30 6 tbase

(ð6Þ

where T30 tbase are in �C and tdpar is dimensionless.The constant j = 1 (mm �C�2 d�1) is included solely toensure Eq. (6) is dimensionally correct and is not usedas an adjustable parameter. Because the degree-dayparameter is frequently set to different values for differ-ent times in the melt season [24], Eq. (6) accounts for theseasonality in melt without adding any additionalparameters.

The daily depth of snowmelt, Md (mm d�1) is com-puted by integrating over the positive portion of thesnow depth p.d.f. (see also Eq. (3))

Md ¼Z 1

Dm

ðminðM s;w� DmÞÞf ðwÞdw ð7Þ
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which reduces melt over the zero and shallow areas ofthe grid cell.

The sub-grid snow parameterization must accountfor situations when new accumulation interrupts melt.In these situations the new snow may fall over the entiregrid cell, and then melt rapidly to leave the original dis-tribution of snow before the accumulation event. Thesesituations have historically been handled by keepingtrack of the new accumulation, and setting snow cov-ered area to 100% until a prescribed percentage of thenew accumulation has melted (e.g., 1,13,16,15). In Lis-ton�s model, new accumulation is removed from totalmelt, Dm, until total melt is zero, then new accumulationis added to total accumulation, Da. This procedureallows the snow covered area to expand and contractover the grid cell during periods when snow is ephem-eral. As noted by Liston [14], the procedure seldom per-mits new accumulation to cover the entire grid cell,which may be unrealistic in some circumstances. How-ever, this procedure preserves the statistical relationsbetween total accumulation and total melt, and SWEand fractional SCA—a feature that is important forthe assimilation strategy outlined in Section 3. In thispaper we use Liston�s method to model transient accu-mulation events.

2.2. Water-balance model

Rain plus melt from the snow model is used as inputto a two-layer water-balance model. A prescribed frac-tion of rain + melt is extracted for direct runoff, Qd

(mm d�1):

Qd ¼ frdir � ðP r þMdÞ ð8ÞW i ¼ P r þMd � Qd ð9Þ

where frdir is a dimensionless model parameter repre-senting the fraction of rain + melt to direct runoff, andWi is the residual water input to the top layer of thewater balance model (mm d�1). As in Section 2.1, Pr isprecipitation in the form of rain and Md is daily snow-melt (both in mm d�1).

The water balance of the top layer is defined by

dW u

dt¼ W i � D� E ð10Þ

where Wu is the water content of the upper layer (mm),D is the drainage from the upper layer to the lower layer(mm d�1), and E is evaporation (mm d�1). The modeluses daily time steps. D is computed as

D ¼W u � smcap; W u > smcap

0; W u 6 smcap

�ð11Þ

where smcap is a model parameter defining the maxi-mum soil moisture capacity (mm). Consequently, drain-age only occurs if Wu > smcap. E is computed as

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Table 1Model parameters and parameter bounds

Parameter Description Lower bound Upper bound

pbias Bias in precipitation 0.5 1.5tsnow Temperature threshold for snowfall �2.0 2.0tmelt Temperature threshold for snowmelt �2.0 2.0tbase Base parameter for the melt factor �20.0 20.0tpowr Exponent for the melt factor 0.1 0.9scvar Coefficient of variation for sub-grid SWE 0.1 0.9etpar Control on the rate of evapotranspiration 0.001 0.1smcap Soil moisture capacity 20.0 300.0frdir Fraction of rain + melt to direct runoff 0.005 0.1fr2gw Fraction of subsurface storage to baseflow 0.005 0.2




RR

EE ¼ PET � W u

smcapð12Þ

with PET as potential evapotranspiration (mm d�1),which is computed using the Hamon empirical formula-tion [11]

PET ¼ etpar � L � qvðsatÞ ð13Þ

where L is the length of daylight (in hours) and qv(sat)

(g m�3) is the saturation absolute humidity, computedusing the mean daily temperature (see [8]). Here etparis an adjustable model parameter.

The water balance of the bottom layer is defined by

dW b

dt¼ D� Qb ð14Þ

Qb ¼ frtgw � W b ð15Þ

where Wb is the storage in the lower layer (mm), Qb isbaseflow (mm d�1), and frtgw is a model parameter thatdefines the fraction of the groundwater storage thatdrains as baseflow per day (d�1).

Daily runoff, Q (mm d�1) is then

Q ¼ Qd þ Qb ð16ÞTable 1 summarizes model parameters and parameterranges.

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CO3. Data assimilation strategy

The data assimilation strategy follows the sub-gridSWE parameterization. With the model structure pre-sented in the previous section, differences between remo-tely sensed and modeled SCA suggests that one or moreof three things is inaccurate: (1) snow accumulation; (2)snowmelt; or (3) the variability in sub-grid SWE (seeFig. 2). Thus, SCA information is used to update totalaccumulation, total melt, and the coefficient of variationparameter. Because snowmelt lingers in soil and ground-water reservoirs, SCA information is also used to updatesoil and groundwater storages. In this approach SWE isnot modified directly, but is altered based on updates to

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of variation parameter, CV (see Eq. (3)).Our implementation of the ensemble Kalman filter

[25] is formalized as follows [10]. Let Xb be the m · n

matrix of ensemble background model states, that is

Xb ¼ ðxb1 ; . . . ; x

bnÞ ð17Þ

where xb1; . . . ; x

bn are vectors of the m model states for

each of the n ensemble members. In this study, n =100 ensemble members are generated for m = 6 modelstates, where the model states are (1) total accumulationdepth, Da; (2) total melt depth, Dm; (3) the coefficient ofvariation in sub-grid SWE, CV; (4) the soil moisturecontent, Wu/smcap; (5) the water stored in the sub-sur-face reservoir, Wb; and (6) the fractional snow coveredarea, SCA. Section 4 outlines methods for producingensemble model simulations.

The model error is estimated directly from the ensem-ble, in which the ensemble mean is used as a index oftruth (the notation below follows [10]. The ensemblemean �xb is defined as

�xb ¼ 1

n

Xni¼1

xbi ð18Þ

The perturbation from the mean for the ith ensemblemember is x0b

i ¼ xbi � �xb, and the ensemble of perturba-

tions is defined as

X0b ¼ ðx0b1 ; . . . ; x

0bn Þ ð19Þ

An estimate of the m · m model error covariance, bPb, is

computed directly from the X 0b ensemble, that is

bPb¼ 1

n� 1X0bX0bT ð20Þ

Having bPb, it is relatively straightforward to update

model states.The update equation is

xai ¼ xb

i þ bKðyi �Hxbi Þ ð21Þ

wherebK ¼ bPbHTðHbPb

HT þ RÞ�1 ð22Þ

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Here xbi is the m-element background vector of model

states for the ith ensemble member, yi is the p-dimen-sional set of observations, H is the m · p operator thatconverts the model state to the observation space, bK isthe Kalman gain, and xa

i is the m-element analysis vectorof model states. R is the p · p observation error covari-ance matrix (Eq. (22)). In this case there is a 1:1 corre-spondence between observations and model states,which avoids the need for interpolation weights in theH matrix. Because only observations of fractionalSCA are available, H = [0,0,0,0,0,1], and HbPb

HT þ R

is a scalar representing the sum of model and observa-tion errors. Accordingly, the Kalman gain bK is a 6-ele-ment column vector in which the covariance betweenfractional SCA and all other model states, bPb

HT, is di-vided by the sum of model and observation errors,HbPb

HT þ R (Eq. (22)).A critical component of the ensemble Kalman filter is

the treatment of observations. In its traditional imple-mentation (Eqs. (21) and (22)), the n elements of y aresampled from a distribution with zero mean and covari-ance R, that is y0i Nð0;RÞ, providing n sets of observa-tions that are used to update each of the n ensemblemembers (see [5]). Whitaker and Hamill [33] introducedan alternative form of the ensemble Kalman filter thatdoes not require perturbed observations. This is theensemble square root Kalman filter and is used in thisstudy. Under this method the ensemble is broken intomean and anomaly portions, which are updatedseparately:

�xa ¼ �xb þ bKðy�H�xbÞ ð23aÞx0a ¼ x0b þ eKHðx0bÞ ð23bÞ

The anomalies (x 0) are updated using a reduced gainðeKÞ, given by

eK ¼ 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R

HbPHT þ R

s !�1 bK ð24Þ

The reduced gain is used because the analysis errorcovariance is underestimated in the standard ensembleKalman filter, which in turn can diminish the spreadof the model ensemble and promote filter divergence[5,33].

For demonstration purposes, R is specified in thisapplication to be 0.05 (i.e., 5% error). Obtaining reliablespatially explicit error estimates from satellite radiancesis a difficult proposition. The biggest challenge involvesunscrambling the radiance from snow from other landcover types in a pixel. These problems are compoundedwhen snow cover is obscured by the vegetation canopy.While the analytical framework presented here is appli-cable for any level of observation error, the Kalmangain clearly depends on the relative magnitude of obser-vation and model error. The 5% error is used only to

demonstrate the applicability of the ensemble Kalmanfilter for SCA assimilation—we acknowledge that fur-ther work on quantifying observational errors is neces-sary to improve filter performance in real-worldapplications.

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4. Probabilistic model simulations

The m · n matrix of ensemble background modelstates, Xb, described in the previous section, is generatedby running the simple water balance model with anensemble of model inputs and a corresponding ensembleof model parameters.

Generating the necessary ensembles requires realdata. Streamflow simulations in this study were pro-duced for Middle Boulder Creek, at Nederland, Colo-rado (United States Geological Survey gageidentification number 06725500). The basin was mod-eled as a single unit (i.e., lumped rather than distributedsimulations). The sub-grid variability in snow is handledusing the p.d.f.s defined in Eqs. (1)–(3).

Middle Boulder Creek is a snowmelt-dominatedmountain basin with a drainage area of 93.76 km2

(Fig. 3). Ensemble precipitation and temperature esti-mates produced for the Lake Eldora SNOTEL site usingthe methods described in Section 4.1 were used as forc-ing data (see also [6,29]). The Lake Eldora SNOTEL islocated within the Middle Boulder Creek drainage(Fig. 3). It is assumed that the ensemble precipitationand temperature estimates for Lake Eldora are represen-tative of the entire Middle Boulder Creek drainage [alarge number of surrounding stations were used to pro-duce the Lake Eldora ensembles [6]]. Middle BoulderCreek has streamflow measurements extending until1995. The probabilistic precipitation and temperatureestimates for Lake Eldora were produced starting in1985. This provides 11 years of data (1985–1995) forwhich streamflow simulations can be compared againstobservations.

4.1. Ensemble model forcings

Ensemble forcing data were produced using the geo-statistical method introduced by Clark and Slater [6].The method is based on locally weighted regression, inwhich spatial attributes from station locations (latitude,longitude, elevation) are used as explanatory variablesto predict spatial variability in precipitation and temper-ature. For each time step, regression models are used toestimate the cumulative distribution function (c.d.f.) ofprecipitation and temperature at a given point (or gridcell).

The precipitation c.d.f. for a given point (and day) iscomputed using locally weighted logistic regression toestimate the probability of precipitation (POP) and

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Elocally weighted linear regression to estimate precipita-tion amounts (PCP), with errors (E) in the amounts esti-mated obtained by locally weighted cross-validatedestimates from interpolations at surrounding stations:

POP ¼ 1

1þ expð�ZTbÞð25Þ

PCP ¼ ZTba ð26Þ

E ¼Pnsta

ista¼1W istaðPCPista � Y istaÞ2Pnstaista¼1W ista

!1=2

ð27Þ

where Z = (1, lat, lon,elev)T is the vector of spatial attri-butes at the target location, b and ba are the regressioncoefficients for the logistic and ordinary least squaresequations respectively, Y is the precipitation amountat a given station, and W is a vector of weights com-puted based on the distance from the target station.The regression coefficients (b and ba) are estimated usinglocal algorithms (see [6] for specific details). The temper-ature c.d.f. is computed in a similar fashion, although itcan be completely defined using Eqs. (26) and (27). Dai-ly precipitation and temperature ensembles are extractedfrom the estimated c.d.f. at the target location, and theensembles are re-ordered to preserve the observedspace–time correlation structure and correlation amongvariables (see [7] for more details). Example forcingensembles are shown for water year 1995 (Fig. 4).

4.2. Ensemble model parameters

The ensemble of model parameters (Table 1) is esti-mated using Monte Carlo Markov Chains [12,32]. Themethod proceeds as follows for a given parameterensemble:

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P1. Randomly select values for all model parametersfrom the feasible parameter space (Table 1), use thisparameter set to simulate streamflow, and computethe difference between model simulations and obser-vations (summarized using the root mean squarederror (RMSE) metric).

2. Randomly select a new parameter set from the feasi-ble parameter space, and compute the RMSE value.

3. Accept the new parameter set if the RMSE value islower than the RMSE value from the old parameterset.

4. Go back to step 2, and continue until convergencecriteria are satisfied or the maximum number of iter-ations reached (400 iterations in this study).

This process was repeated for 100 ensemble members,producing an ensemble of 100 parameter sets that areconsistent with the data. To ensure parameter valueswere not ‘‘over-fit’’ to the noise in any given forcingensemble, the RMSE was calculated from streamflowsimulations forced with 10 randomly selected forcingensembles. Fig. 5 shows the RMSE value for eachparameter set for the first 100 iterations (later iterationsare in gray shades), and the probability distributionfunction for each parameter after 400 iterations. Therange of parameter values (Table 1) was deliberatelyset to be quite large, partly to test the capability of theMonte Carlo Markov Chain method to select realisticparameter values. Consistent with known precipitationunder-catch problems, the method selects pbias valuesabove 1.0 (Fig. 5). Also, note that while some parametervalues clearly produce more accurate streamflow simula-tions, a large number of different parameter sets are con-sistent with the data.

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Fig. 4. Example forcing ensembles for the coordinates of Lake Eldora SNOTEL site for water year 1995. The dots are station observations (not usedto construct the ensembles), and the shading depicts percentiles from the ensemble (0.05, 0.25, 0.45, 0.55, 0.75, 0.95). While these ensembles wereconstructed for a specific point, we assume for the purposes of our demonstration that these forcing ensembles are valid for the entire Middle BoulderCreek drainage.

Fig. 5. Root mean squared error for different model parameter sets, as obtained from the Monte Carlo Markov chains. The plus signs depictparameter sets for the first 100 iterations (later iterations are shaded gray), and the lines depict the probability distribution functions for eachparameter after 400 iterations. See Table 1 for parameter definitions.




4.3. Implementation

Having estimated an ensemble of model forcings andan ensemble of model parameters, it is now possible to

produce ensemble model simulations that concurrentlyaccount for uncertainties in model forcings and uncer-tainties in the choice of model parameters. To achievethis, we constructed 100 model ensembles in which we

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Fig. 6. Ensemble simulations of streamflow for Middle Boulder Creek for the period 1984–1995 (every second year is shown). The dots arestreamflow observations, and the shading depicts percentiles from the ensemble (0.05, 0.25, 0.45, 0.55, 0.75, 0.95).




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randomly selected (without replacement) one forcingensemble and one parameter ensemble. This is done onlyto avoid the extra computational expense of 10,000ensembles (i.e., all possible combinations of forcingand parameter ensembles). Fig. 6 illustrates probabilisticstreamflow simulations for every second year in the per-iod 1985–1995. The dots are streamflow observationsfrom the Middle Boulder Creek gage at Nederland,and the shading depicts percentiles from the simulatedensemble (i.e., 0.05, 0.25, 0.45, 0.55, 0.75, 0.95). Theprobabilistic estimates encompass the observationsfairly well.
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For assimilation of SCA to have any effect on othermodel state variables, there must be significant covari-ance between SCA and the other model states (e.g.,see Eq. (22)). To assess covariance across model states,we present in Fig. 7 the correlation (the normalizedcovariance) between SCA and the other model statevariables for each of the 11 years in our study period.The coefficient of variation parameter has generally neg-ative correlations, which implies that ground is exposedearlier (less snow coverage) when snow cover is morevariable. Positive correlations for total accumulationimply that more snow accumulation is associated with

Tmore snow coverage, and the negative correlations withtotal melt imply that more melt is associated with lesssnow coverage. The correlations between SCA andSWE are positive (note that SWE is not used as a statevariable but is derived from total accumulation andtotal melt).

Turning to the state variables in the water-balancemodel: correlations are much stronger with the ground-water storage than with soil moisture. This occursbecause the soil moisture reservoir is an imperviousbucket, in which all water spills to groundwater storageonce the bucket has reached capacity (defined by theparameter smcap; Eq. (11)). The soil moisture reservoiris filled early in the melt season, which means fractionalsoil moisture is almost constant across model ensembles.Groundwater storage is represented as a bucket of infi-nite size, with baseflow dependent on the amount ofwater in the bucket (see Eqs. (14) and (15)). The negativecorrelations at the start of the melt season mean thatincreased snow coverage is associated with decreasedgroundwater storage—that is, groundwater storage islower when melt is delayed. The positive correlationsat the end of the melt season mean that increased snowcoverage is associated with higher groundwater stor-age—in this case higher snow coverage implies a length-ening of the snow season and maintenance of the highgroundwater levels typical of snowmelt periods. Updat-ing state variables in the water-balance model is essential

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Fig. 7. Correlations among model states for each of the 11 years in the study period. Each year is represented by a separate line.




to keep track of the water in the basin and improve sim-ulations of streamflow.
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Synthetic ‘‘identical twin’’ experiments (e.g., [25–27])were used to evaluate the potential of the proposed dataassimilation strategy. In twin experiments the model isfirst run for the period of record using the best possibleforcing data and parameter sets. This serves as the‘‘true’’ solution and is meant to represent nature [27].The model is then run with degraded forcing and param-eter sets. These simulations, termed ‘‘control’’ or ‘‘openloop’’ model runs, are meant to represent the current sit-uation in which models cannot replicate nature. Finally,model states from the ‘‘true’’ solution are assimilatedinto the ‘‘control’’ runs. These simulations are termedthe ‘‘assimilation’’ model runs. Potential benefits of dataassimilation can be evaluated by comparing output fromthe ‘‘control’’ and ‘‘assimilation’’ runs against modeloutput from the ‘‘true’’ model run. The advantage oftwin experiments is that there are synthetic data for allmodel states, for which observations are oftenunavailable.

For this study, the twin experiment proceeds slightlydifferently. It is assumed that each parameter/forcingcombination is an equally likely description of nature.The first parameter set and forcing ensemble are selectedand the model is run for the 11-year period (1985–1995).This model run is assumed to be truth. Next, the modelis run for the other 99-parameter/forcing ensembles.These ensemble simulations are considered as the con-trol run. Finally, the fractional SCA is assimilated fromthe first ensemble into the control ensemble. This pro-cess is repeated for all 100-ensemble members (i.e., eachensemble member is eventually used as ‘‘truth’’). Thisexperiment provides 1100 synthetic years to evaluatethe effect of the assimilation strategy (11 years * 100ensemble members). Assimilation started on March1st, to focus on the spring period when snow cover isvariable.

Based on the correlations in Fig. 7, assimilation ofSCA should have a dramatic effect on model simulationsof snowpack and the basin water balance. Fig. 8 pre-sents example results for one twin experiment for wateryear 1995, showing the control ensemble (left column),the assimilation ensemble (middle column), and the con-trol and assimilation p.d.f.s for an example day near theend of June (right column). Fig. 8 shows that the ‘‘true’’

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Fig. 8. Example ensemble simulations of model states and streamflow for water year 1995. The left column shows results for the ‘‘control’’ ensemble,the middle column shows results for the ‘‘assimilation’’ ensemble, and the right column shows example p.d.f.s for July 1st (light line is control, darkline is assimilation). Synthetic observations from the ‘‘truth’’ simulation are shown as dots, and the shading depicts percentiles from the ensemble(0.05, 0.25, 0.45, 0.55, 0.75, 0.95). See text for further details.




fractional SCA is lower than most of the control ensem-bles (top row); that is, bare ground is exposed earlier inthe �true’’ simulation. Early exposure of bare groundmay be due to higher variability in sub-grid SWE (i.e.,a higher CV parameter), which implies more shallowsnow. Consistent with this notion, assimilating frac-

tional SCA from the ‘‘true’’ simulation results in anincrease in the CV parameter (second column ofFig. 8). Increases in the CV parameter are restricted tothe month of June when SCA is variable. Bare groundcan also be exposed earlier if there is less total snowaccumulation in nature than in the control simulations.

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The assimilation of fractional SCA from the ‘‘true’’ sim-ulation results in a decrease in total snow accumulation(Fig. 8). Finally, bare ground can be exposed earlier ifthere is more total melt in nature than in the controlsimulations. In this example, there are only minorchanges in total melt, which may occur as a result ofthe covariability between melt and other model statevariables. The ensemble streamflow simulations (bottomrow of Fig. 8) have a better correspondence to the syn-thetic observations after fractional SCA is assimilated.

Fig. 9 illustrates seasonal cycles in the mean ensemblevariance in simulated streamflow (top row), and theRMSE of the ensemble mean streamflow (bottomrow), averaged over the 1100 synthetic water years. Boththe variance in the ensemble streamflow simulations andthe RMSE is slightly lower in the assimilation runs (lightline) than in the control simulations (dark line). Thereductions in variance and errors are most pronouncedduring the months of June and July. The relatively smalleffect of SCA assimilation occurs for two reasons. First,a substantial proportion of spring streamflow occursbefore any bare ground is exposed, meaning that assim-ilating fractional SCA information will always have onlya limited effect on ensemble simulations of streamflow.

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Fig. 9. Mean ensemble variance (top row) and the root mean squarederror (bottom row) averaged over 1100 synthetic water years. Thecontrol run is depicted by the dark line, and the assimilation run isdepicted by the light line.

Second, the transition from conditions when 100% ofthe basin is snow covered to conditions when 100% ofthe basin is snow free can occur fairly quickly (e.g.,occasionally within a 2-week period), providing a veryshort temporal window for using fractional SCAinformation.

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7. Summary and discussion

An alternative analytical framework for assimilatingfractional snow covered area (SCA) information inhydrologic and land-surface models was presented.The assimilation method uses SCA information in anensemble Kalman filter to alter the sub-grid probabilitydistribution function (p.d.f.) of snow (defined in termsof total accumulation, total melt, and the variability ofSWE). It was demonstrated that assimilating SCA infor-mation effectively modifies the sub-grid distribution ofSWE, as well as the basin water balance. Assimilationof SCA also modifies the ensemble simulations ofstreamflow.

The summary statistics presented in this paper dem-onstrate that for the basin examined in this study assim-ilating SCA information provides only minor increasesin model accuracy. This occurs because a substantialproportion of spring streamflow occurs before any bareground is exposed, and because the transition from con-ditions when 100% of the basin is snow covered to con-ditions when 100% of the basin is snow free often occursfairly quickly. While these two factors do constrainapplications of satellite SCA information in basinsworldwide, the relative importance of these two factorswill be basin-dependent. We expect satellite SCA infor-mation to be most useful in basins where snow cover isephemeral. The improvements in streamflow simulationdemonstrated in this study indicate that fractional SCAis an important source of independent information thatis useful as part of an integrated data assimilationstrategy.

The results presented in this paper are based on syn-thetic experiments and are still a few steps away fromreal-world applications. Andreadis and Lettenmaier [2]recently completed a pilot study assessing the use ofthe ensemble Kalman filter to assimilate remotely sensedSCA information in the Snake River basin, Idaho, USA.They constructed forcing ensembles by perturbing pre-cipitation inputs with fixed error limits (25%), and spec-ified observation errors at 10%. Comparison with pointobservations showed the filter significantly reducederrors in model simulations of SWE in the ablation sea-son. Consistent with the results in our study, reductionin errors were minimal in the accumulation season whensnow cover is always close to 100%, and the reduction inerrors were much lower at high elevations, where snowcover is also more continuous.

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There are a number of ways to improve filter perfor-mance. First, it is necessary to account for temporal per-sistence in the model updates (e.g., [25]). Not doing socan be viewed as analogous to repeatedly assimilatingthe same observation in a single model time step—thiswill reduce the variance of the ensemble, meaning thatthe filter will increasingly favor the model predictions.Hence, the filter will operate in a sub-optimal manner.Slater and Clark [29] provide a simple remedy to thisissue. Another area that needs more attention is theassumption of normally distributed errors. In manyhydrologic systems errors are certainly not normally dis-tributed, which can also lead to sub-optimal modelupdates. New approaches are emerging that can addressmore complex error structures (e.g., [34,20]). These issuesnotwithstanding, the biggest challenge in effectivelyapplying data assimilation techniques lies in quantifyingthe model and observation error statistics. Too oftenmodel and observation errors are either prescribed, orcrudely parameterized. It is necessary to attack this errorestimation problem with fervor and gusto.

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Acknowledgments

This work was primarily supported by the NationalWeather Service Office of Hydrologic Development(Award NWS4620014). Partial support was also pro-vided by the NOAA GAPP Program (AwardNA16GP2806), and the NOAA RISA Program (AwardNA17RJ1229). This support is gratefully acknowledged.Three anonymous reviewers provided useful suggestions.

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References

[1] Anderson EA. National Weather Service river Forecast System—snow accumulation and ablation model. NOAA technical mem-orandum NWS HYDRO-17. Silver Spring, MD: US Depart-ment of Commerce; 1973.

[2] Andreadis KM, Lettenmaier DP. Assimilating remotely-sensedsnow observations into a macroscale hydrology model. AdvWater Resour, in press.

[3] Barrett AP, Leavesley GH, Viger RL, Nolin AW, Clark MP. Acomparison of satellite-derived and modeled snow-covered areafor a mountain drainage basin. In: Owe M, Brubaker K, Ritchie J,Rango A, editors. Remote sensing and hydrology 2000 (Proceed-ing of a symposium held at Santa Fe, New Mexico, USA, April2000). IAHS Publ. No. 267, 2001. p. 569–73.

[4] Barrett AP. Integrating remotely sensed snow cover with adistributed hydrologic model. Hydrol Process, submitted forpublication.

[5] Burgers G, van Leeuwen PJ, Evensen G. Analysis scheme in theensemble Kalman filter. Mon Wea Rev 1998;126(6):1719–24.

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[6] Clark MP, Slater AG. Probabilistic quantitative precipitationestimation in complex terrain. J Hydrometeorol, in press.

[7] Clark MP, Gangopadhyay S, Hay LE, Rajagopalan B, Wilby RL.The Schaake Shuffle: a method for reconstructing space–timevariability in forecasted precipitation and temperature fields. JHydrometeorol 2004;5:243–62.

[8] Dingman LS. Physical hydrology. New Jersey: Prentice-Hall;1994. 575 p.

[9] Gomez-Landsea E, Rango A. Operational snowmelt runoffforecasting in the Spanish Pyrenees using the snowmelt runoffmodel. Hydrol Process 2002;16:1583–91.

[10] Hamill TM. Ensemble-based atmospheric data assimilation. In:Palmer TN, Hagedorn R, editors. Predictability of weather andclimate. Cambridge Press, 2005, in press.

[11] Hamon WR. Estimating potential evapotranspiration. J HydraulDiv, Proc Am Soc Civil Eng 1961;87:107–20.

[12] Kuczera G, Parent E. Monte Carlo assessment of parameteruncertainty in conceptual catchment models: the Metropolisalgorithm. J Hydrol 1998;211:69–85.

[13] Leavesley GH, Lichty RW, Troutman BM, Saindon LG. Precip-itation-runoff modeling system: users manual. US GeologicalSurvey Water Resources Investigation Report, 83-4238, 1983.

[14] Liston GE. Representing subgrid snow cover heterogeneities inregional and global models. J Climate 2004;17:1381–97.

[15] Luce CH, Tarboton DG. The application of depletion curves forparameterization of sub-grid variability of snow. Hydrol Process2004;18:1409–22.

[16] Luce CH, Tarboton DG, Cooley CR. Sub-grid parameterizationof snow distribution for an energy and mass balance snow covermodel. Hydrol Process 1999;13:1921–33.

[17] Martinec J. Limitation in hydrological interpretations of the snowcoverage. Nordic Hydrol 1980;11:209–20.

[18] Martinec J, Rango A, Roberts R. Snowmelt runoff model (SRM)users manual. Geographica Bernensia, Department of Geogra-phy, University of Bern; 1994. p. 65.

[19] Mcguire M, Lettenmaier DP. Use of satellite data for streamflowand reservoir storage forecasts in the Snake River basin, ID. JWater Resour Planning Manage, submitted for publication.

[20] Moradkhani H, Hsu K, Gupta H, Sorooshian S. Uncertaintyassessment of hydrologic model states and parameters: sequentialdata assimilation using the particle filter. Water Resour Res2005;41:W05012. doi:10.1029/2004WR003604.

[21] Rango A. Operational applications of satellite snow coverobservations. Water Resour Bull 1980;16:1066–73.

[22] Rango A. Progress in snow hydrology remote sensing research.IEEE Trans Geosci Remote Sensing 1986;GE-24:47–53.

[23] Rango A. Spaceborne remote sensing for snow hydrologyapplications. Hydrol Sci J 1996;41:477–93.

[24] Rango A, Martinec J. Revisiting the degree-day method forsnowmelt computations. Water Resour Bull 1995;31:657–69.

[25] Reichle RH, McLaughlin DB, Entekhabi D. Hydrologic dataassimilation with the ensemble Kalman filter. Mon Wea Rev2002;130(1):103–14.

[26] Reichle RH, Walker JP, Koster RD, Houser PR. Extended versusensemble Kalman filtering for land data assimilation. J Hydro-meteorol 2002;3(6):728–40.

[27] Reichle RH, Koster RD. Assessing the impact of horizontal errorcorrelations in background fields on soil moisture estimation. JHydrometeorol 2003;4(6):1229–42.

[28] Rodell M, Houser PR. Updating a land surface model withMODIS-derived snow cover. J Hydrometeorol 2004;5:1064–75.

[29] Slater AG, Clark MP. Snow data assimilation via an ensembleKalman filter. J Hydrometeorol, in press.

[30] SMHI. IHMS: integrated hydrologic modeling system manual,version 4.0. Norrkoping, Sweden: Swedish Meteorological andHydrological Institute; 1996.

http://dx.doi.org/10.1029/2004WR003604

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[31] Turpin O, Ferguson R, Johansson B. Use of remote sensing to testand update simulated snow cover in hydrologic models. HydrolProcess 1999;13:2067–77.

[32] Vrugt JA, Gupta HV, Bouten W, Sorooshian S. A shuffledcomplex evolution metropolis algorithm for optimization and

UN

CO

RR

EC

828829

uncertainty assessment of hydrologic model parameters. WaterResour Res 2003;39(8). Art. No. 1201.

[33] Whitaker JS, Hamill TM. Ensemble data assimilation withoutperturbed observations. Mon Wea Rev 2002;130(7):1913–24.

[34] Zupanski M. Maximum likelihood ensemble filter. Theoreticalaspects. Mon Wea Rev 2005;133:1710–26.

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